Ƙarfin Lissafi na Haɗin Kai: Tsarin Da ke Haɗa Rashin Yanki da Lissafi na Tushen Ma'auni

Teburin Abubuwan Ciki

• 1.1 Gabatarwa & Bayyani
• 1.2 Tsarin Tsakiya: Lissafi na Tushen Ma'auni
• 1.3 Ma'anar Ƙarfin Lissafi na Haɗin Kai
• 2.1 Haɗi zuwa Rashin Yanki na Ƙididdiga
• 2.2 GHZ da CHSH a matsayin Jihohin Albarkatu Mafi Kyau
• 3.1 Tsarin Fasaha & Tsarin Lissafi
• 3.2 Tasirin Gwaji & Sakamako
• 4.1 Tsarin Bincike: Nazarin Lamari Wanda Ba Code ba
• 4.2 Aikace-aikace na Gaba & Hanyoyin Bincike
• 5. Nassoshi
• 6. Ra'ayin Mai Bincike: Fahimtar Tsakiya, Kwararar Hankali, Ƙarfi & Kurakurai, Fahimtoci Masu Aiki

1.1 Gabatarwa & Bayyani

Wannan aikin da Anders da Browne suka yi yana bincika wata muhimmiyar tambaya a mahadar bayanin ƙididdiga da ka'idar lissafi: Menene ƙarfin lissafi na asali na haɗin kai? Bayan wucewa daga aiwatarwa ta musamman kamar na'urar lissafi ta ƙididdiga ta hanya ɗaya, marubutan sun gina tsari gabaɗaya don auna daidai yadda albarkatun haɗin kai—waɗanda ake samun su ta hanyar ma'auni—za su iya haɓaka ƙarfin kwamfutar sarrafa gargajiya. Babban binciken da ya ba da mamaki shi ne haɗin kai kai tsaye tsakanin keta ka'idojin gaskiya na yanki (rashin yanki na ƙididdiga) da amfanin lissafi na yanayin haɗin kai a cikin wannan tsari.

1.2 Tsarin Tsakiya: Lissafi na Tushen Ma'auni

Marubutan sun ayyana samfuri gabaɗaya wanda ya ƙunshi sassa biyu:

1. Albarkatu na Haɗin Kai na Bangarori Da Yawa: Ƙungiyar bangarori (misali, qubits) waɗanda ba sa sadarwa yayin lissafi. Kowane bangare yana karɓar shigarwar gargajiya (ɗaya daga cikin zaɓuɓɓukan $k$) daga kwamfutar sarrafawa kuma yana mayar da sakamako na gargajiya (ɗaya daga cikin sakamakon $l$). Haɗin kai a cikin sakamakonsu an riga an ƙaddara su ta hanyar yanayinsu na raba ko tarihi.
2. Kwamfutar Sarrafa Gargajiya: Na'urar da ke da ƙayyadaddun ƙarfin lissafi (misali, ƙayyadaddun ƙwaƙwalwar ajiya, iyakataccen zurfin da'ira) wanda ke tsara lissafin. Tana aika shigarwa zuwa ga bangarorin albarkatu, tana karɓar sakamakonsu, kuma tana aiwatar da sarrafa gargajiya, mai yuwuwa ta amfani da sakamakon don zaɓar shigarwar gaba daidai.

Babban ƙuntatawa shi ne cewa kowane bangaren albarkatu ana hulɗa da shi kawai sau ɗaya yayin wani lissafi. Wannan tsari yana cire ƙididdiga, yana mai da hankali kawai kan halayen shigarwa-fitowa na gargajiya waɗanda haɗin kai marasa gargajiya ke sauƙaƙa.

1.3 Ma'anar Ƙarfin Lissafi na Haɗin Kai

"Ƙarfin lissafi" na albarkatu mai haɗin kai an ayyana shi dangane da kwamfutar sarrafa gargajiya. Albarkatu yana ba da ƙarfin lissafi idan, ta hanyar amfani da shi, kwamfutar sarrafawa za ta iya warware matsalar lissafi wadda ba za ta iya warwarewa da kanta ba. Wannan yana kaiwa ga ra'ayin jihohin albarkatu don lissafi na gargajiya na tushen ma'auni (MBCC). Marubutan suna neman siffanta waɗanne tsarin haɗin kai (waɗanda aka ƙirƙira ta hanyar rarraba yuwuwar sharadi $P(\text{sakamako}|\text{shigarwa})$) su ne albarkatu masu amfani.

2.1 Haɗi zuwa Rashin Yanki na ƙididdiga

Takardar ta kafa haɗin kai mai zurfi: haɗin kai waɗanda suka keta rashin daidaiton Bell (don haka ba su da samfurin ɓoyayyen ma'auni na yanki) su ne ainihin waɗanda za su iya zama albarkatun lissafi marasa ban sha'awa a cikin tsarin MBCC. Wannan saboda rashin yanki yana ba da damar albarkatu don ƙirƙirar dogaro tsakanin sakamakon ma'auni wanda kwamfutar gargajiya, tana aiki a ƙarƙashin ƙuntatawa na yanki, ba za ta iya samar da su da kanta ba.

2.2 GHZ da CHSH a matsayin Jihohin Albarkatu Mafi Kyau

Abin mamaki, sanannun tsarin rashin yanki sun fito a matsayin misalai mafi kyau:

• Yanayin Greenberger-Horne-Zeilinger (GHZ): Haɗin kai a cikin sabani na GHZ yana ba da albarkatu don warware takamaiman matsalar lissafi da aka rarraba (mai alaƙa da "wasan GHZ" ko lissafin XOR na bit na daidaito) wadda kwamfutar gargajiya ba za ta iya warwarewa ba tare da sadarwa tsakanin bangarorin ba.
• Rashin Daidaiton Clauser-Horne-Shimony-Holt (CHSH): Haɗin kai na bangarori biyu waɗanda suka fi keta rashin daidaiton CHSH ($S = 2\sqrt{2}$) sun dace da albarkatu wanda ke ba da fa'ida a cikin lissafin aikin Boolean, yana fi kowace albarkatu mai haɗin kai na gargajiya da aka iyakance da $S \leq 2$.
• Akwatin Popescu-Rohrlich (PR): Wannan akwatin ka'idar da ba shi da yanki sosai (amma ba shi da alamar sigina) yana wakiltar albarkatu mai kyau wanda zai ba da fa'idar lissafi mafi girma a cikin wannan samfurin, yana warware matsaloli tare da tabbacin da ba zai yiwu ga albarkatun gargajiya ba.

Wannan sakamakon ya sake tsara waɗannan abubuwan asali na ƙididdiga ba kawai a matsayin gwaje-gwaje na gaskiyar yanki ba, amma a matsayin ma'auni na amfanin lissafi.

3.1 Tsarin Fasaha & Tsarin Lissafi

Ana iya tsara tsarin ta amfani da rarraba yuwuwar sharadi. Albarkatu $R$ an ayyana shi ta hanyar saitin yuwuwar $P(a_1, a_2, ..., a_n | x_1, x_2, ..., x_n)$, inda $x_i$ shine shigarwa ga bangare $i$ kuma $a_i$ shine sakamakonsa. Albarkatu yana rashin sigina idan:

$\sum_{a_i} P(a_1,...,a_n|x_1,...,x_n)$ ya kasance mai zaman kansa daga $x_i$ ga duk $i$.

An ƙayyade lissafi ta hanyar aiki $f$ wanda dole ne kwamfutar sarrafawa ta ƙididdige shi, mai yuwuwa ta amfani da dabarun daidaitawa dangane da sakamakon tsaka-tsaki daga albarkatu. Ana tantance ƙarfin lissafi ta hanyar kwatanta yuwuwar nasara ko ingancin lissafin $f$ tare da albarkatu $R$ da ba tare da shi ba (ko kawai tare da haɗin kai na gargajiya).

3.2 Tasirin Gwaji & Sakamako

Duk da yake takardar ta kasance ta ka'ida, tasirinta ana iya gwada shi. Gwaji da zai nuna MBCC zai ƙunshi:

1. Saitawa: Shirya yanayin haɗin kai na bangarori da yawa (misali, yanayin GHZ na photons).
2. Sarrafawa: Kwamfutar gargajiya (misali, FPGA) wanda ke yanke shawarar tushen ma'auni (shigarwa $x_i$) ga kowane na'urar gano photon.
3. Lissafi: Kwamfutar tana karɓar sakamakon ganowa ($a_i$) kuma tana amfani da su, bin tsarin algorithm da aka riga aka ƙayyade, don ƙididdige ƙimar aiki (misali, daidaiton shigarwar da aka rarraba).
4. Sakamako: Ƙimar nasarar wannan lissafi zai wuce matsakaicin da za a iya samu idan an maye gurbin hanyoyin samar da photon da na'urorin ƙirƙira lambobi na gargajiya tare da raba bazuwar, waɗanda aka iyakance ta hanyar rashin daidaiton Bell. "Ginshiƙi" zai nuna yuwuwar nasara akan axis-y da ƙarfin haɗin kai (misali, ƙimar CHSH $S$) akan axis-x, tare da bayyanannen bakin kofa a iyakar gargajiya ($S=2$).

4.1 Tsarin Bincike: Nazarin Lamari Wanda Ba Code ba

Lamari: Wasan CHSH a matsayin Aikin Lissafi.

Aiki: Bangarori biyu da suka rabu, Alice da Bob, suna karɓar bit ɗin bazuwar mai zaman kansa $x$ da $y$ (bi da bi) daga Kwamfutar Sarrafawa. Manufarsu ita ce samar da fitarwa $a$ da $b$ irin $a \oplus b = x \cdot y$ (XOR ya yi daidai da AND).

Dabarar Gargajiya (tare da raba bazuwar): Matsakaicin yuwuwar nasara shine $75\%$ ($3/4$). Wannan shine iyakar gargajiya, daidai da $S \leq 2$.

Dabarar Ƙididdiga (ta amfani da qubits masu haɗin kai): Ta hanyar raba nau'i-nau'i masu haɗin kai da ma'auni a cikin tushe da aka zaɓa bisa ga $x$ da $y$, za su iya cimma yuwuwar nasara na $\cos^2(\pi/8) \approx 85.4\%$. Wannan yayi daidai da iyakar Tsirelson $S = 2\sqrt{2}$.

Bincike: A cikin tsarin MBCC, Kwamfutar Sarrafawa tana ciyar da $x$ da $y$ a matsayin shigarwa ga albarkatun ƙididdiga (nau'i-nau'i masu haɗin kai). Ana mayar da fitarwa $a$ da $b$. Sannan kwamfutar tana ƙididdige $a \oplus b$, wanda zai yi daidai da $x \cdot y$ tare da yuwuwar $\sim85.4\%$. Wannan aikin lissafi ne—ƙididdige aikin AND da aka rarraba ta hanyar XOR—wanda kwamfutar sarrafawa ke aiwatarwa da aminci fiye da yadda za ta iya amfani da kowace albarkatu mai haɗin kai na gargajiya. Haɗin kai mara yanki shine man lissafin.

4.2 Aikace-aikace na Gaba & Hanyoyin Bincike

• Ma'auni na Na'urorin Ƙididdiga: Wannan aikin yana ba da sabon ma'ana mai aiki don keta rashin daidaiton Bell—ba kawai a matsayin gwaje-gwaje na kimiyyar lissafi ba, amma a matsayin daidaitawa na amfanin lissafi. Ƙimar CHSH ta na'ura tana nuna ƙarfinta kai tsaye a matsayin albarkatu don wasu abubuwan asali na lissafi da aka rarraba.
• Ƙirƙirar Algorithms na Haɗin Ƙididdiga da Gargajiya: Fahimtar waɗanne haɗin kai na ƙididdiga ke warware waɗanne matsalolin gargajiya na iya jagorantar ƙirƙirar algorithms na haɗin gwiwa inda ƙaramin na'urar ƙididdiga ke aiki a matsayin "na'urar haɓaka haɗin kai" ga babban injin sarrafa gargajiya, ra'ayin da ke samun karbuwa a zamanin NISQ.
• Rubutun Sirri da Tabbatarwa: Haɗin kai na asali tsakanin rashin yanki da ƙarfin lissafi yana ƙarfafa ka'idojin rubutun sirri masu zaman kanta na na'ura. Idan na'urar mai nisa za ta iya taimakawa warware matsalar da ta wahala ta gargajiya ta hanyar MBCC, tana tabbatar da kasancewar albarkatun ƙididdiga na gaske.
• Bayan Ka'idar Ƙididdiga: Tsarin yana ba da damar bincika ka'idoji tare da haɗin kai mafi ƙarfi fiye da ƙididdiga (kamar akwatunan PR) da sakamakon lissafinsu, yana sanar da neman ƙa'idodin zahiri waɗanda ke iyakance ƙididdiga.

5. Nassoshi

1. R. Raussendorf da H. J. Briegel, "Kwamfutar Ƙididdiga Ta Hanya Guda," Phys. Rev. Lett. 86, 5188 (2001).
2. D. E. Browne da H. J. Briegel, "Lissafi na Ƙididdiga Ta Hanya Guda," a cikin Lectures on Quantum Information, Wiley-VCH (2006).
3. M. A. Nielsen, "Lissafi na Yanayin Cluster," Rep. Math. Phys. 57, 147 (2006).
4. N. Brunner et al., "Rashin Yanki na Bell," Rev. Mod. Phys. 86, 419 (2014).
5. J. F. Clauser et al., "Gwaji da aka Tsara don Gwada Ka'idojin ɓoyayyun Ma'auni na Yanki," Phys. Rev. Lett. 23, 880 (1969).
6. D. M. Greenberger et al., "Ka'idar Bell ba tare da rashin daidaito ba," Am. J. Phys. 58, 1131 (1990).
7. S. Popescu da D. Rohrlich, "Rashin Yanki na Ƙididdiga a matsayin Axiom," Found. Phys. 24, 379 (1994).
8. IBM Quantum, "Menene ma'aunin ƙarar ƙididdiga?" [Kan layi]. Ana samuwa: https://www.ibm.com/quantum/computing/volume/

6. Ra'ayin Mai Bincike: Fahimtar Tsakiya, Kwararar Hankali, Ƙarfi & Kurakurai, Fahimtoci Masu Aiki

Fahimtar Tsakiya: Anders da Browne sun ba da babban ra'ayi ta hanyar sake tsara rashin yanki na ƙididdiga—wanda ya daɗe yana batun muhawara na asali—a matsayin albarkatun lissafi mai ƙima. Babban ra'ayinsu shi ne cewa "sihiri" na haɗin kai na ƙididdiga ba kawai game da kin gaskiyar yanki ba ne; kuɗi ne mai canzawa wanda za a iya kashewa don warware takamaiman, ƙayyadaddun matsalolin gargajiya waɗanda ba su isa ga haɗin kai na gargajiya ba. Wannan yana haɗa wani babban rashi tsakanin tushen ƙididdiga na zahiri da kimiyyar bayanin ƙididdiga da aka yi amfani da ita.

Kwararar Hankali: Hujjar an gina ta da kyau. 1) Taƙaice: Cire ƙididdiga don ayyana samfurin gabaɗaya na "kwamfutar gargajiya + akwatuna masu haɗin kai baƙaƙe" (MBCC). 2) Ƙididdiga: Ayyana ƙarfin lissafi a matsayin fa'ida dangane da kwamfutar gargajiya kaɗai. 3) Haɗa: Tabbatar cewa albarkatun da ke ba da irin wannan fa'ida su ne ainihin waɗanda suka keta rashin daidaiton Bell. 4) Misalta: Nuna cewa misalan al'ada (GHZ, CHSH, akwatin PR) ba kawai abubuwan sha'awa ba ne amma albarkatu mafi kyau a cikin wannan kasuwar lissafi. Kwararar daga taƙaice zuwa misalan takamaiman yana da ban sha'awa.